Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.2 The Derivative as a Function - Exercises - Page 117: 90



Work Step by Step

We have the derivative: $$ f^{\prime}(x)=3x^{2} $$ Then at $x=a, m=f^{\prime}(a)=3a^{2}$. Hence, the tangent line is: $$ \begin{aligned} \frac{y-y_{1}}{x-x_{1}} &=m \\ \frac{y-a^n}{x-a} &=3a^{2} \\ y &=3a^{2} (x-a)+a^3 \end{aligned} $$ Since the tangent line intersects with the $x-$ axis at $x=0,$ then $Q$ has coordinates $(a-\frac{a}{3},0), R$ has coordinates $(a,0)$, and the subtangent is $$ a-\left(a-\frac{a}{3}\right)=\frac{a}{3} $$
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